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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2005 Dutch Mathematical Olympiad
2005 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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Numbers in an array
Consider an array of numbers of size
8
×
8
8 \times 8
8
×
8
. Each of the numbers in the array equals 1 or -1. "Doing a move" means that you pick any number in the array and you change the sign of all numbers which are in the same row or column as the number you picked. (This includes changing the sign of the "chosen" number itself.) Prove that one can transform any given array into an array containing numbers +1 only by performing this kind of moves repeatedly.
4
1
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Trapezoid and midpoints
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
A
B
>
C
D
AB > CD
A
B
>
C
D
. Prove that the line passing through
A
C
∩
B
D
AC \cap BD
A
C
∩
B
D
and
A
D
∩
B
C
AD \cap BC
A
D
∩
BC
passes through the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
.
3
1
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Smallest possible value of m
Let
a
1
,
a
2
,
a
3
,
a
4
,
a
5
a_1,a_2,a_3,a_4,a_5
a
1
,
a
2
,
a
3
,
a
4
,
a
5
be distinct real numbers. Consider all sums of the form
a
i
+
a
j
a_i + a_j
a
i
+
a
j
where
i
,
j
∈
{
1
,
2
,
3
,
4
,
5
}
i,j \in \{1,2,3,4,5\}
i
,
j
∈
{
1
,
2
,
3
,
4
,
5
}
and
i
≠
j
i \neq j
i
=
j
. Let
m
m
m
be the number of distinct numbers among these sums. What is the smallest possible value of
m
m
m
?
2
1
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Regular dodecagon
Let
P
1
P
2
P
3
…
P
12
P_1P_2P_3\dots P_{12}
P
1
P
2
P
3
…
P
12
be a regular dodecagon. Show that
∣
P
1
P
2
∣
2
+
∣
P
1
P
4
∣
2
+
∣
P
1
P
6
∣
2
+
∣
P
1
P
8
∣
2
+
∣
P
1
P
10
∣
2
+
∣
P
1
P
12
∣
2
\left|P_1P_2\right|^2 + \left|P_1P_4\right|^2 + \left|P_1P_6\right|^2 + \left|P_1P_8\right|^2 + \left|P_1P_{10}\right|^2 + \left|P_1P_{12}\right|^2
∣
P
1
P
2
∣
2
+
∣
P
1
P
4
∣
2
+
∣
P
1
P
6
∣
2
+
∣
P
1
P
8
∣
2
+
∣
P
1
P
10
∣
2
+
∣
P
1
P
12
∣
2
is equal to
∣
P
1
P
3
∣
2
+
∣
P
1
P
5
∣
2
+
∣
P
1
P
7
∣
2
+
∣
P
1
P
9
∣
2
+
∣
P
1
P
11
∣
2
.
\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.
∣
P
1
P
3
∣
2
+
∣
P
1
P
5
∣
2
+
∣
P
1
P
7
∣
2
+
∣
P
1
P
9
∣
2
+
∣
P
1
P
11
∣
2
.
1
1
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Divisible by 5
In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5?